Should he pay zakaah on the expenses that his father gives to him?

Firstly: 

Zakaah is due on cash if you have full possession of the
nisaab (minimum threshold) and one full hijri year has passed. 

The expenses that a father gives to his son is something that
the son takes full possession of and disposes of as he wishes, so that is
subject to zakaah. 

Secondly: 

It is well known that cash in the form of bank notes was not
known at the time of the Prophet (peace and blessings of Allaah be upon
him), but the scholars have ruled that it is subject to zakaah by analogy
with gold and silver. 

The nisaab for gold is 85 grams, and the nisaab for silver is
595 grams. 

If the cash reaches the value of the nisaab for gold or the
value of the nisaab for silver, then it has reached the minimum threshold. 

As silver is of a lower price nowadays, the nisaab for cash
may be worked out on the basis of the nisaab for silver, because that is
more on the safe side and is more beneficial to the poor. 

Based on the price of silver today –  12 Rabee’ al-Aakhir
1428 AH/29 April 2007 CE – the nisaab for cash is approximately 1093 Saudi
Riyals. 

If you own this amount of money and one full hijri year has
passed since you acquired this amount and it has not dipped below the nisaab
during this year, then zakaah must be paid on it, at a rate of 2.5 %. 

But if the cash that you possessed dipped below the nisaab
during the year, then zakaah does not have to be paid on it, until it
reaches the nisaab again and you start to reckon a new year from the time it
reached the nisaab. 

If the amount by which it dipped below the nisaab was small,
then in order to be on the safe side you may pay zakaah, and carry on
counting the year. That is because the price of silver fluctuates and does
not remain constant throughout the year.  

We should not forget to commend your keenness and concern
with regard to zakaah, even though you are talking about pocket money that
you take from your father, but you are paying attention to Allaah’s right to
it and have asked about the Islamic ruling on it at a time when many rich
people neglect this pillar of Islam and do not acknowledge Allaah’s rights
over their wealth or spend a lot or a little of it, and they spend their
days accumulating and hoarding and seeking to acquire more, but at the
reckoning on the Day of Resurrection their wealth will be a cause of loss
and regret. 

Allaah says (interpretation of the meaning): 

“And those who hoard up gold and silver (Al‑Kanz: the
money, the Zakaah of which has not been paid) and spend them not in the way
of Allaah, announce unto them a painful torment.

35. On the Day when that (Al‑Kanz: money, gold and silver,
the Zakaah of which has not been paid) will be heated in the fire of Hell
and with it will be branded their foreheads, their flanks, and their backs,
(and it will be said unto them:) ‘This is the treasure which you hoarded for
yourselves. Now taste of what you used to hoard’”

[al-Tawbah 9:34-35]

We ask Allaah to bless your wealth for you and to grant you
abundant halaal provision. 

And Allaah knows best.